doc. Ing. Pavel enovsk, Ph.D. Modeling of Decision processes Balance models Wassily Leontief Prof. Wassily Leontief (1906-1999) Since 1932 Harward University 1973 Nobel prize for economy Model of general equilibrium Input-output analysis Input output model quantitative economic technique that represents the interdependencies between different branches of a national economy or different regional economies Leontief used the model to quantify impact of the World War II on the Economy of USA Method is usable also to not strictly economical problems like evaluationg impact of large disruptions of infrastructure See Hazus https://www.fema.gov/hazus Input data large problem The tables describing communication between branches of the economy are required Its formulation is complicated if attempted, it is available 5 7 years after Data represent snapshot of the economy of that time Not all countries even try Where to next? Since snapshot of the whole economy is
complicated it is possible to simplify by analyzing smaller chunks of data, i.e. data of the region or a single company This simplification leads to Balance models Other approach - analyze only the connections, do not try to quantify it -> modelling of interconnections in critical infrastructure Balance models - purpose Definition They are formalizing connections between basic materials, energetical requirements etc. of modeled system. Realization of these connections is required for transformation of the inputs to outputs. Model form: systems of (usually linear) equations How do we create model Identification of the elements of the model Identification of connection between them Definition of parameters and variables of the model Formulation and quantification of the model Model solution 1. Identification of model elements Products Semi-finished product Raw materials fuel energy wrapping Elements of model Products and semi-products Raw materials, fuel, energy No.
Name No. Name 1 Green RJ 1 Steam 2 Green RP 2 Electrical energy 3 Green khaki 3 Mirbam oil 4 Green BJ 4 Antrachinon
5 Zele BK 5 Benzoylchlorid 6 Zele BK 6 Oleum 7 Zele bx 7 Sulphuric acid 8 Olive green B 8 Sulphate 9 Olive green A 9 nitrogen
10 Benzimidin 10 Amoniac water 11 Diantrimid 11 Sulphide soda 12 Benzatron 12 Kortamol 13 Antrachinonsulfonan 13 Lye soda 2. Identification of connections Definition of parameters and variables Semi-products and products
yj production of j-product (or semi-product) for usage outside of the model xj overall production of product j without looking at where it will go xij usage of product i for production of xj units of the product j Definition of parameters and variables Material inputs si amouth of the matial needed to fullfill needs of the customers xij = aij * xj si = sij * xj where aij specific need of semiproduct i sij spec. Need of material inputs for unit of semiproduct/product Formulation and quantification of the model Realization in two phases Model of connection between the products Balance model (may be supported by capacity model) Modeling of inter-product connection Overall amount of produced final products equals the sum of customer needs (of the products) and its usage for production of the different products xk = xk1 + xk2 + + xkn or we can use xij from deffiniton of material inputs xk = ak1x1 + ak2x2 + + aknxn Balance of inter-product connections x1 = a11x1 + a12x2 + + a1nxn
x2 = a21x1 + a22x2 + + a2nxn xn = an1x1 + an2x2 + + annxn Balance model for amount of material input sp = sp1 x1 + sp2x2 + + spnxn s1 = s11x1 + s12x2 + + s1nxn s2 = s21x1 + s22x2 + + s2nxn sm = sm1x1 + sm2x2 + + smnxn Solution using IT Usage of matrixes more appropriate A square matrix of type (n, n) of specific usage of the semi-products aij. Majority of the matrix elements will equal to zero (semi-product is not used to produce product xi) x vector of overall production of the product xj. y vector of production going outside of the model Solution using IT B matrix of type (m, n) consists of specific usage coef. of the material inputs needed to produce products and semi-products sij s vector of overall needs of materials si we search for Equations using the matrix x = y + Ax Model of usage of material and energetical inputs Bx = s Capacity model Used for estimation of feasibility of the plan from capacity point of view Cx = f where C rectangle matrix of type (r, n), where r is number of
machines, production lines (usually in hours needed for production of the unit) f vector of the type (r,1) searched for requirements on length of use of the machines Specific needs of semiproducts Type Inputs: energy, raw materials, semi-products Specific need unit/t green Green BJ Green BB Kortamol electricity Green BK Green BX Kortamol electricity 0,025 0,978 0,03 2,1 0,081 0,2 0,15 1,45 Green khaki Model of inter-product connections
Balance of semi-products Product 1 green RJ Balance equation x 1 = y1 2 green RP x 2 = y2 3 green khaki x 3 = y3 4 green BJ x4 = y4 + 0,025x1 5 green BK x5 = y5 + 0,81x2 + 0,02x3 Balance of raw materials and energy Balance equations steam s1 = 1,2x8 + 17,5x9 + 11x10 + 32,5x11 + 28x12 + 24,5x13 electricity s2 = 2,1x1 + 0,95x2 + 1,45x3 + 1,45x4 + 3x2 + 1,3x6 + 3x7 + 1,45x8 + 1,6x9 + 1,2x10 + 1,2x11 + 1x12 + 0,51x13 Model expressed as matrix x y x1
y1 x2 x3 y2 y3 x4 x5 x6 y4 y5 y6 x7 y 7 x8 y8 x9 x10 x11 y9 y10 y11 x12 x13 y12 y13 A x 0 0
0 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 Solving balance model x = [E - A]-1y Or [E - A]-1 = U x = Uy Where E unit matric U square inverse matrix to matrix [E-A]-1. Its elements uij are overall specific needs of semi-rpoducts for needs of production Resulting matrix U 1 0 0 1 0 0 0 0 0 0
1,1071 1,0244 0,208 0 0 1,04 1,04 0,0139 0,0164 0,4518 0,5565 0,5565 0 0,0053 0,4475 0,451 0,0915 0 0 0,4576 0,4576 0,0092 0,0108 0,2982 0,3673 0,3673 0 0,0035 0,34987 0 0,66 0 1 0 0,3222 0,325 0,0659 0 0 0,3295 0,3295 0 0,3168 0 0,72 0 1 Application to Continuity management Balance model allowed us to understands the structure of the production By applying constrains to various components to simulate impact of the accident we gain capability to simulate impact to the production as whole Interpretation of the results can be used as basis for planning for accident
IIM input output interoperability model Mathematically very similar original: x = Ax + c x = the Production Vector in individual economic sectors; A = the Consumption Matrix that enables the definition of links between sectors; c = the External Demand Vector for the outside of the modelled system IIM Simplifies the model only connections between the sectors are established, so x <0; 1> To model cascading effects x = 0 -> represents complete break down of connected CI sector x = 1 -> no effect (full functionality on CI sector) IIM Dependency index (row summation) The dependency index expresses the robustness of an infrastructure with respect to the inoperability of the other infrastructures it represents the maximum inoperability in infrastructure i when every other infrastructure is fully inoperable Dependency index interpretation . The greater the decrease in gi (when gi < 1), the better infrastructure i can maintain its working capabilities Influence gain (column summation) A large value of rj means that the inoperability of jth sector induces significant degradation to the system
Hazus different approach Vulnerability curve to hazard evaluation im IIM Impact on production Impact on other regions Damage to facility
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